A few of us posted at the same time. +1 to what others posted.
The barrel is lead fouled ... When I hold the barrel up to the light I can see the lead sticking to the barrel in places and keep working to get it out.
Check your barrel bore diameter and bullet diameter to see they are sized for each other. My M&P barrel is .451" and I shoot .452" Missouri Bullets. I get little or no leading depending on the powder/charge I use. As to removing lead, after soaking the barrel, running copper scrubber wrapped old bore brush works well for me.
will lead also build up in the chamber as this would be harder to clean than the barrel.
No, most of fouling/gas cut lead deposit will be forward of the chamber as when the primer ignites the powder, case neck will expand against the chamber and make a tight seal. You may get "some" fouling buildup over time and I just use some solvent to soak and rotate a brush around the chamber to clean.
4.0 grains of Bullseye and 200gr SWC lead cast bullets.
I do not know what the hardness of your bullet is, but 4.0 gr of Bullseye for 200 gr bullet is a light load and may not be bumping the base enough to seal against the bore (obturation). If you do not get a good seal when the primer ignites the powder, you will get gas cutting and increased leading at chamber end and down. Try test loads at higher charges to see if the leading decreases. I shoot MBC 18 BHN bullets at mid-high range load data and 12 BHN bullets at start-mid range. Also, longer OAL that will reliably feed/chamber from the magazine will also reduce the "jump" gap and give less time for the hot gas to escape around the bullet, further reducing the gas cutting of the bullet.
powder build up from the bullseye seems exessive. Any suggestions for cleaning or different loads?
Most powders will burn dirtier at lower charges and cleaner at higher charges as you increase the chamber pressure. I would try a higher powder charge instead of "better" cleaning methods.
Different loads? I use Bullseye for 9mm/45ACP also and it burns dirtier than W231/HP38 at comparable loads.